1. Functions of more than one variable Math 2000 Winter 2013 semester

2. Math 1000 and 1001 review In first-year calculus we focus on functions of one variable: y = f(x)f: R  R We work with these functions to introduce concepts like: – Limits and continuity – Derivatives – Integrals Many (most?) real-life functions depend on more than one thing, however!

3. Example: Temperature map What does the temperature depend on? – Location (x,y) – Altitude (z)

4. Example: Temperature map What does the temperature depend on? – Location (x,y) – Altitude (z) – Time (t)

5. Example: Temperature map So, we could represent temperature as a function of four variables: T = f(x,y,z,t) f: R 4  R For every value of (x,y,z,t), there is a single corresponding value of T. What does this mean for our concept of things like derivatives and integrals?

6. Partial derivatives Recall that the derivative measures the rate of change of a function. In 1D we have: – What does this mean for a multivariate function? – What might we mean when we ask, “how fast is the temperature changing?” Change with respect to position? Change with respect to time? Change with respect to some combination of these?

7. Multiple integration Recall that the integral can give us the average value of a function over some interval. The same is true for a multivariate function: – e.g. The average temperature of a surface region at some fixed time: – e.g. The average temperature of that region over some time period:

8. Visualizing multivariate functions For the rest of the course we will look at extending first-year calculus concepts to functions of more than one variable. An important first step is to consider how we visualize a such a function. For functions of 2 dimensions, f: R 2  R, there are several ways. We will look at some examples.

9. E.g. This function of two variables defines a surface in 3D: Visualizing functions of 2 variables

10. E.g. Another way of visualizing 2D functions is with a contour map:

11. E.g. Finally, we can also fix the value of one of the variables and plot the trace of the function with respect to the other variable. Visualizing functions of 2 variables

12. E.g. Visualizing functions of 2 variables

13. E.g. Visualizing functions of 2 variables

14. E.g. Note that this function is defined only if The domain of the function is Visualizing functions of 2 variables

15. Visualizing in higher dimensions For a function of 3 variables, we can look at level surfaces, or traces through the function.

16. Even though we can “visualize” a function of 3 variables, it is common to look at 2D slices through the function instead. Visualizing in higher dimensions

17. Unfortunately we can only see in 3 dimensions! We can only look at part of a function of more than 3 variables at one time.

18. Visualizing in higher dimensions Unfortunately we can only see in 3 dimensions! We can only look at part of a function of more than 3 variables at one time.

19. Visualizing in higher dimensions Unfortunately we can only see in 3 dimensions! We can only look at part of a function of more than 3 variables at one time.